Take 1/2 cup from the top layer of coconut milk and place in a large saucepan over medium heat. Heat to bubbling, then stir in red curry paste. Reduce heat and simmer for 5 minutes. Add remaining coconut milk and broth; increase heat to medium and bring to a boil. Add lemon grass, chicken, ginger and fish sauce; return to a boil. Reduce heat and simmer, uncovered, until chicken is cooked through, about 8 minutes.

Grate zest from the lime, then squeeze the juice. Add lime zest, 1 tbsp lime juice and baby spinach to the soup; simmer for 5 minutes. Remove the lemon grass and discard. Taste and add lime juice and fish sauce, if desired.